Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties

524 Pages·1999·14.59 MB·English

by  Giorgio Ausiello

Complexity
and
Approximation
Springer-Verlag Berlin Heidelberg GmbH
G. Ausiello P. Crescenzi G. Gambosi
V. Kann A. Marchetti -Spaccamela M. Protasi
Complexity
and Approximation
Combinatorial Optimization Problems
and Their Approximability Properties
With 69 Figures and 4 Tables
Springer
Giorgio Ausiello Pierluigi Crescenzi
Alberto Marchetti-Spaccamela Dipartimento di Sistemi e Informatica
Dipartimento di Informatica Universita degli Studi di Firenze
e Sistemistica Via C. Lombroso 6117
Universita di Roma "La Sapienza" 1-50134 Florence, Italy
Via Salaria 113, 1-00198 Rome, Italy
Viggo Kann
Giorgio Gambosi NADA, Department of Numerical
Marco Protasi t Analysis and Computing Science
Dipartimento di Matematica KTH, Royal Institute of Technology
Universita di Roma "Tor Vergata" SE-10044 Stockholm, Sweden
Via della Ricerca Scientifica
1-00133 Rome, Italy
Cover picture "What happened in the night" by J. Nesetfil and J. Naceradsky
Second corrected printing 2003
Library of Congress Cataloging-in-Publication Data
Complexity and approximation: combinatorial optimization problems and
their approximability properties/G. Ausiello ... let al.l.
p. cm.
Includes bibliographical references and index.
ISBN 978-3-642-63581-6 ISBN 978-3-642-58412-1 (eBook)
DOI 10.1007/978-3-642-58412-1
1. Combinatorial optimization. 2. Computational complexity.
3. Computer algorithms. I. Ausiello, G. (Giorgio), 1941-
QA402.S.CSSS 1999
S19.3-dc21 99-40936
CIP
ACM Subject Classification (1998): F.2, G.1.2, G.1.6, G.2, G.3, GA
1991 Mathematics Subject Classification: 05-01,90-01
Additional material to this book can be downloaded from http://extras.springer.com
ISBN 978-3-642-63581-6
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© Springer-Verlag Berlin Heidelberg 1999
Softcover reprint of the hardcover 1st edition 1999
The use of general descriptive names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from
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Typesetting: Camera-ready by the authors
Design: design + production GmbH, Heidelberg
Printed on acid-free paper SPIN 10885020 06/3142SR - 5 4 3 2 1 0
To our dear colleague
and friend Marco Protasi, in memoriam .
.. .A nd soonest our best men with thee doe goe,
rest of their bones, and soules deliverie.
JOHN DONNE
(j. . 9L
P.L.e.
(j.(j.
o/.!l( .
.9L.!M.-S.
To Gabriele, Igor, Irene, and Sara
To Giorgia and Nicole
To Benedetta
To Elon
To Salvatore
To Davide and Sara
I will tell you plainly
all that you would like to know,
not weaving riddles,
but in simple language,
since it is right
to speak openly to friends.
AESCHYLUS,
Prometheus bound, 609-611
Contents
1 The Complexity of Optimization Problems 1
1.1 Analysis of algorithms and complexity of problems 2
1.1.1 Complexity analysis of computer programs 3
1.1.2 Upper and lower bounds on the complexity of
problems . . . . . . . . . . . . . 8
1.2 Complexity classes of decision problems . 9
1.2.1 The class NP . . . . . . . . 12
1.3 Reducibility among problems . . . . 17
1.3.1 Karp and Turing reducibility 17
1.3.2 NP-complete problems . . . 21
1.4 Complexity of optimization problems 22
1.4.1 Optimization problems . . . . 22
1.4.2 PO and NPO problems . . . . 26
1.4.3 NP-hard optimization problems 29
1.4.4 Optimization problems and evaluation problems 31
1.5 Exercises ...... 33
1.6 Bibliographical notes . . . . . . . . . . . . . . 36
2 Design Techniques for Approximation Algorithms 39
2.1 The greedy method . . . . . . . . . . . . . . . ... . 40. .
2.1.1 Greedy algorithm for the knapsack problem . . .. 41
2.1.2 Greedy algorithm for the independent set problem 43
2.1.3 Greedy algorithm for the salesperson problem . .. 47
Table of contents
2.2 Sequential algorithms for partitioning problems 50
2.2.1 Scheduling jobs on identical machines . 51
2.2.2 Sequential algorithms for bin packing 54
2.2.3 Sequential algorithms for the graph coloring problem 58
2.3 Local search. . . . . . . . . . . . . . . . . . . . . . . .. 61
2.3.1 Local search algorithms for the cut problem . . .. 62
2.3.2 Local search algorithms for the salesperson problem 64
2.4 Linear programming based algorithms . . . . . . . 65
2.4.1 Rounding the solution of a linear program . 66
2.4.2 Primal-dual algorithms 67
2.5 Dynamic programming . . . . . . . . . . . . . . . 69
2.6 Randomized algorithms. . . . . . . . . . . . . . . 74
2.7 Approaches to the approximate solution of problems 76
2.7.1 Performance guarantee: chapters 3 and 4 76
2.7.2 Randomized algorithms: chapter 5 . 77
2.7.3 Probabilistic analysis: chapter 9 77
2.7.4 Heuristics: chapter 10 78
2.7.5 Final remarks 79
2.8 Exercises ..... . 79
2.9 Bibliographical notes 83
3 Approximation Classes 87
3.1 Approximate solutions with guaranteed performance 88
3.1.1 Absolute approximation . . . . . . . . . . . 88
3.1.2 Relative approximation . . . . . . . . . . . . 90
3.1.3 Approximability and non-approximability of TSP. 94
3.1.4 Limits to approximability: The gap technique 100
3.2 Polynomial-time approximation schemes. 102
3.2.1 The class PTAS . . . . . . . . . . . . . 105
3.2.2 APX versus PTAS .......... . 110
3.3 Fully polynomial-time approximation schemes III
3.3.1 The class FPTAS . . . . . . . . . . . III
3.3.2 The variable partitioning technique .. 112
3.3.3 Negative results for the class FPTAS . . 113
3.3.4 Strong NP-completeness and pseudo-polynomiality 114
3.4 Exercises ...... 116
3.5 Bibliographical notes . . . . . . . . . . . . . . 119
4 Input-Dependent and Asymptotic Approximation 123
4.1 Between APX and NPO. . . . . . . . . . . . . 124
4.l.l Approximating the set cover problem 124
4.l.2 Approximating the graph coloring problem 127
viii
Table of contents
4.1.3 Approximating the minimum multi-cut problem. 129
4.2 Between APX and PTAS . . . . . . . . . . . . . . 139
4.2.1 Approximating the edge coloring problem . 139
4.2.2 Approximating the bin packing problem. 143
4.3 Exercises ...... 148
4.4 Bibliographical notes . . . . . . . . 150
5 Approximation through Randomization 153
5.1 Randomized algorithms for weighted vertex cover. 154
5.2 Randomized algorithms for weighted satisfiability . 157
5.2.1 A new randomized approximation algorithm 157
5.2.2 A 4/3-approximation randomized algorithm. 160
5.3 Algorithms based on semidefinite programming . . . 162
5.3.1 Improved algorithms for weighted 2-satisfiability 167
5.4 The method of the conditional probabilities 168
5.5 Exercises .... . . 171
5.6 Bibliographical notes . . . . . . . . . . 173
6 NP,PCP and Non-approximability Results 175
6.1 Formal complexity theory. . . . . . . . 175
6.1.1 Turing machines . . . . . . . . 175
6.1.2 Deterministic Turing machines . 178
6.1.3 Nondeterministic Turing machines. 180
6.1.4 Time and space complexity. . . . . 181
6.1.5 NP-completeness and Cook-Levin theorem 184
6.2 Oracles ... . ..... . ... 188
6.2.1 Oracle Turing machines 189
6.3 The PCP model . . . . . . . . . 190
6.3.1 Membership proofs. . . 190
6.3.2 Probabilistic Turing machines 191
6.3.3 Verifiers and PCP. . . . . . . 193
6.3.4 A different view of NP .... 194
6.4 Using PCP to prove non-approximability results. 195
6.4.1 The maximum satisfiability problem . 196
6.4.2 The maximum clique problem 198
6.5 Exercises ...... 200
6.6 Bibliographical notes 204
7 The PCP theorem 207
7.1 Transparent long proofs . 208
7.1.1 Linear functions 210
7.1.2 Arithmetization. 214
ix
Table of contents
7.1.3 The first PCP result . . . 218
7.2 Almost transparent short proofs. 221
7.2.1 Low-degree polynomials 222
7.2.2 Arithmetization (revisited) 231
7.2.3 The second PCP result 238
7.3 The final proof .. .. .. .. . 239
7.3.1 Normal form verifiers . 241
7.3.2 The composition lemma 245
7.4 Exercises .. . .. . 248
7.5 Bibliographical notes . . . . . . 249
8 Approximation Preserving Reductions 253
8.1 The World of NPO Problems 254
8.2 AP-reducibility . . . . . . . 256
8.2.1 Comph:~te problems. 261
8.3 NPO-completeness . . . . . 261
8.3.1 Other NPO-complete problems. 265
8.3.2 Completeness in exp-APX . . . 265
8.4 APX-completeness . . . . . . . . . . . 266
8.4.1 Other APX-complete problems . 270
8.5 Exercises ...... 281
8.6 Bibliographical notes . . . . . . . . . . 283
9 Probabilistic analysis of approximation algorithms 287
9.1 Introduction............ . ....... 288
9.1.1 Goals of probabilistic analysis . . . . . . 289
9.2 Techniques for the probabilistic analysis of algorithms 291
9.2.1 Conditioning in the analysis of algorithms. 291
9.2.2 The first and the second moment methods . . 293
9.2.3 Convergence of random variables ... . .. 294
9.3 Probabilistic analysis and multiprocessor scheduling 296
9.4 Probabilistic analysis and bin packing . . . 298
9.5 Probabilistic analysis and maximum clique 302
9.6 Probabilistic analysis and graph coloring . 311
9.7 Probabilistic analysis and Euclidean TSP . 312
9.8 Exercises . . .... 316
9.9 Bibliographical notes 318
10 Heuristic methods 321
10.1 Types of heuristics 322
10.2 Construction heuristics 325
10.3 Local search heuristics 329
x